The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces (original) (raw)
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2022
We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincar\'e inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Mal\'y, Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that is approximable from above and below by continuous functions. We also show that for each finL1(partialOmega),f\in L^1(\partial\Omega),finL1(partialOmega), there is a least gradient function in Omega\OmegaOmega whose trace agrees with fff at points of continuity of fff, and so we obtain existence of solutions for boundary data which is continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan, who constructed an L1L^1L1-function on the unit circle which has no least gradient solution in the unit disk in mathbbR2.\mathbb{R}^2.mathbbR2. Modifying the example of Spradlin...
Lebesgue points and capacities via the boxing inequality in metric spaces
Indiana University Mathematics Journal, 2008
The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.
Variational problems concerning length distances in metric spaces
arXiv (Cornell University), 2023
Given a locally compact, complete metric space (X, D) and an open set Ω ⊆ X, we study the class of length distances d on Ω that are bounded from above and below by fixed multiples of the ambient distance D. More precisely, we prove that the uniform convergence on compact sets of distances in this class is equivalent to the Γ-convergence of several associated variational problems. Along the way, we fix some oversights appearing in the previous literature. The rest of the introduction is subdivided as follows: first we describe more in details our main results, then we compare them with the previous works in this field. Statement of results. Let (X, D) be a locally compact, complete metric space, Ω ⊆ X an open set, and α > 1 a given constant. Our main object of study is the class D α (Ω) of all length distances on Ω verifying α −1 D d αD, cf. with Definition 2.4. We point out that the continuous extension d of d to the closure Ω × Ω is a geodesic distance onΩ verifying α −1 D d αD, see Lemma 2.7. We will consider three different functionals associated with any given distance d ∈ D α (Ω): • The length functional Ld, which is defined on the space of Lipschitz curves from [0, 1] to the closureΩ, the latter being equipped with the topology of uniform convergence; see (2.1). • The 'optimal-transport-type' functional J d , which is defined as J d (µ) := d(x, y) dµ(x, y) for every finite Borel measure µ 0 on Ω × Ω; see (2.2). The domain of J d is equipped with the weak * topology.
First order Poincaré inequalities in metric measure spaces
Annales Academiae Scientiarum Fennicae Mathematica, 2013
We study a generalization of classical Poincaré inequalities, and study conditions that link such an inequality with the first order calculus of functions in the metric measure space setting when the measure is doubling and the metric is complete. The first order calculus considered in this paper is based on the approach of the upper gradient notion of Heinonen and Koskela [HeKo]. We show that under a Vitali type condition on the BMO-Poincaré type inequality of Franchi, Pérez and Wheeden [FPW], the metric measure space should also support a p-Poincaré inequality for some 1 ≤ p < ∞, and that under weaker assumptions, the metric measure space supports an ∞-Poincaré inequality in the sense of [DJS].
Lebesgue points and capacities via boxing inequality in metric spaces
Indiana University Mathematics Journal, 2008
The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.
Covering theorems, inequalities on metric spaces and applications to PDE’s
Mathematische Annalen, 2008
We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.